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The Theory of Induced Representations in Field Theory

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The Theory of Induced Representations in Field Theory QMW{PH{95{? hep-th/yymmnnn The Theory of Induced Representations in Field Theory Jose´ M. Figueroa-O'Farrill ABSTRACT These are some notes on Mackey's theory of induced representations. They are based on lectures at the ITP in Stony Brook in the Spring of 1987. They were intended as an exercise in the theory of homogeneous spaces (as principal bundles). They were never meant for distribution. x1 Introduction The theory of group representations is by no means a closed chapter in mathematics. Although for some large classes of groups all representations are more or less completely classified, this is not the case for all groups. The theory of induced representations is a method of obtaining representations of a topological group starting from a representation of a subgroup. The classic example and one of fundamental importance in physics is the Wigner construc- tion of representations of the Poincar´egroup. Later Mackey systematized this construction and made it applicable to a large class of groups. The method of induced representations appears geometrically very natural when expressed in the context of (homogeneous) vector bundles over a coset manifold. In fact, the representation of a group G induced from a represen- tation of a subgroup H will be decomposed as a \direct integral" indexed by the elements of the space of cosets G=H. The induced representation of G will be carried by the completion of a suitable subspace of the space of sections through a given vector bundle over G=H. In x2 we review the basic notions about coset manifolds emphasizing their connections with principal fibre bundles. In doing so we find it instructive to review the basic concepts associated with fibre bundles. The reader familiar with this should only scan this section to familiarize him/herself with the notation. In x3 we give the construction of induced representations of G carried by the space of sections through a homogeneous vector bundle or equivalently by functions defined on the group subject to an equivariance property. In x4 we discuss the unitarity of the induced representation and introduce the concept of a multiplier representation. In x5 we restrict ourselves to a special class of Lie groups (containing the spacetime groups) for which the method of induced representations yields all irreducible unitary representations and we shall give a method by which to obtain them. The method will be a generalization of Wigner's \little group" method. This usually gives us a representation in a different space than the one we would like to use and, moreover, covariance under H transformations is often not manifest. Hence in x6 we introduce the so-called covariant functions and present a method to relate induced representations on one coset | where the representations are irreducible | to another coset where the representations are more useful. This will be of fundamental importance since the constraints we must impose to guarantee irreducibility in the second coset are | in the case of the Poincar´e group | nothing but the free field equations. Finally in x7 we look at some familiar examples in detail. The following is a list of references | by no means complete | from which these notes have evolved: { 2 { [BR] A. O. Barut and R. R¸aczka, Theory of Group Representations and Appli- cations. [H1] R. Hermann, Lie Groups for Physicists. [H2] R. Hermann, Fourier Analysis on Groups and Partial Wave Analysis. [Ma] G. W. Mackey, Induced Representations of Groups and Quantum Mechan- ics. [NO] U. H. Niederer and L. O'Raifeartaigh, Forschritte der Physik, 22, 111-158 (1974). [NW] P. van Nieuwenhuizen and P. West, Chapter 10 of forthcoming book. [We] S. Weinberg, Quantum Theory of Massless Particles. x2 General facts about Coset Spaces Given any group G and a subgroup H we have a natural equivalence re- lation between elements of G. We say that two elements g; g0 are equivalent (g ∼ g0) if g0 = g · h for some h 2 H. This partitions G into equivalence classes called (left) H cosets and the collection of all such cosets is denoted G=H. There is a canonical map G ! G=H sending each group element to the coset containing it. We write this map as g 7! [g]. If G is a topological group, then all cosets are homeomorphic (as subspaces of G) and, indeed, if G is a Lie group and H a Lie subgroup then G=H can be given a differentiable structure such that the canonical map is smooth and furthermore G ! G=H is a fibra- tion of G by H. We now review some basic notions about fibre bundles. We shall, in view of the applications we have in mind, work in the differentiable category. All our spaces will be differentiable manifolds and all our maps will be smooth. A fibre bundle is a generalization of the cartesian product. It consists of two differentiable manifolds E; B called the total space and base respectively with a smooth surjection π: E ! B such that every point p 2 B has a neigh- bourhood U such that its preimage in E looks like a cartesian product. That ' −1 is, there exists a diffeomorphism 'U : U ×F −!π (U) giving local coordinates to E, where F is a topological space called the typical fibre. Hence locally E looks like B×F although generally this is not the case in the large. If this is the case then the bundle is called trivial. We often omit mention of the fibre when this is understood and refer to a fibre bundle simply as E−!π B. Now cover B with a collection of trivializing neighbourhoods fUig and let p 2 Ui \ Uj. Let 'i and 'j be the coordinate maps associated with Ui and Uj, respectively. Then the preimage in E of Ui \Uj can be given coordinates in two ways: either by 'i or by 'j. The change of coordinates corresponds to a reparametrization { 3 { −1 ◦ \ × of the fibre. More explicitly the map 'i 'j mapping (Ui Uj) F to itself breaks up as id × gij where the gij: Ui \ Uj ! DiffF are called the transition functions. Generally the transition functions take values in a more manageable sub- group of DiffF and we get different kinds of fibre bundles depending on this subgroup as well as on particular properties of F . For instance if F is a vector space (say, V) and the transition functions take values in GL(V) | i.e.the linear structure of V is preserved | then the bundle is called a vector bun- dle. Now let F be a Lie group (say, H) and let there be a smooth free action of H on E on the right (for definiteness) which preserves the fibred structure | i.e.the action of H does not move a point of E away from its fibre | or, symbolically, denoting the right action of h 2 H by Rh, such that π ◦ Rh = π. In this case we say that E is fibred by H and the base B is diffeomorphic to the space of orbits E=H. The transition functions then take values in H and the whole structure is called a principal fibre bundle. In particular when E is a Lie group G and the action of H is just right multiplication then the base can be identified with the space of left cosets G=H. A way to give coordinates to the coset space G=H is to use the coordinates already present in G. The way to do this is to choose smoothly a representative for each coset. That is, for every coset p 2 G=H choose an element of σ(p) 2 G such that [σ(p)] = p. In fibre bundle language such a map is called a section. Given a bundle E−!π B any (smooth) map σ: B ! E such that π ◦ σ = id is called a (smooth) section. Whereas a vector bundle has many sections (e.g.the zero section assigning to every point in the base the zero vector in the fibre) the existence of a global section in a principal fibre bundle is equivalent to triviality. Indeed we show that given a global section we can construct a ∼ diffeomorphism E = B × H. Given any point e 2 E map it to (π(e); h) where e = Rh (σ (π(e))). Because the action of H is transitive on each fibre and free such an h always exist and is unique. Moreover the map is smooth because the action of H and π are both smooth. Conversely a trivial principal fibre bundle admits a global section. In fact, a global section in this case is nothing but the graph of a smooth function B ! H. We shall, for calculational and notational convenience, work with a global section. This, although true for the case of contractible bases such as Minkowski spacetime, is not always possible. However if the base contains a subspace M whose complement is of measure zero (with respect to a suitable measure on the base) and such that the bundle over M is trivial we can safely (i.e.for the purposes we have in mind) assume that the bundle is trivial. We shall see later how this arises. Given a principal fibre bundle E−!π B with fibre H and a representation { 4 { D: H ! GL(V) of H we can construct a vector bundle with fibre V called an associated vector bundle. The construction runs as follows. Consider the cartesian product E × V and define on it an equivalence relation as follows −1 (e; v) ∼ (Rh(e);D(h ) · v) for all h 2 H.
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